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Research Article, Res Rep Math Vol: 2 Issue: 1

Extending the Applicability of an Ulm-Newton-like Method under Generalized Conditions in Banach Space

Ioannis K. Argyros1* and Santhosh George2

Department of Forestry and Biodiversity, Tripura University, Suryamaninagar, Agartala, India

*Corresponding Author : Ioannis K. Argyros Department of Mathematical Sciences, Cameron University, Lawton, OK 73505, USA, E-mail: [email protected]

Received: November 02, 2017 Accepted: January 15, 2018 Published: February 10, 2018

Citation: Argyros IK, George S (2018) Extending the Applicability of an Ulm-Newton-like Method under Generalized Conditions in Banach Space. Res Rep Math 2:1

Abstract

The aim of this paper is to extend the applicability of an Ulm-Newtonlike method for approximating a solution of a nonlinear equation in a Ba-nach space setting. The su cient local convergence conditions are weaker than in earlier works leading to a larger radius of convergence and more precise error estimates on the distances involved. Numerical examples are also provided in this study. AMS Subject Classi cation: 65H10, 65G99, 65J15,49M15.

Keywords: Ulm’s method; Banach space; local / semi-local conver-gence

Introduction

In this study we are concerned with the problem of approximating a locally unique solution x of equation

F (x) = 0; (1.1)

where, F is a Frechet{di erentiable operator de ned on a convex subset of a Banach space B1 with values in a Banach space B2.

A large number of problems in applied mathematics and also in engineering are solved by nding the solutions of certain equations. For example, dynamic systems are mathematically modeled by di erence or di erential equations, and their solutions usually represent the states of the systems. For the sake of sim-plicity, assume that a time{invariant system is driven by the equation x = R(x), for some suitable operator R, where x is the state. Then the equilibrium states are determined by solving equation (1.1). Similar equations are used in the case of discrete systems. The unknowns of engineering equations can be func-tions (di erence, di erential, and integral equations), vectors (systems of linear or nonlinear algebraic equations), or real or complex numbers (single algebraic equations with single unknowns). Except in special cases, the most commonly used solution methods are iterative{when starting from one or several initial approximations a sequence is constructed that converges to a solution of the equation. Iteration methods are also applied for solving optimization problems. In such cases, the iteration sequences converge to an optimal solution of the problem at hand. Since all of these methods have the same recursive structure, they can be introduced and discussed in a general framework [1-12].

Moser [13] proposed the following Ulm’s-like method for generating a sequence fxng approximating x :

equation

Method (1.2) is useful when the derivative F 0(xn) is not continuously invertible (as in the case of small divisors [1-8,10,11,13-15]). Moser studied the semi-localpconvergence of method (1.2) and showed that the order of convergence is 1 + 2 if F 0(x?) 2 L(B2; B1p). However, the order of convergence is faster than the Secant method (i.e. 2). The quadratic convergence can be obtained if one uses Ulm’s method [14,15]

equation

equation

The semi-local convergence of method (1.3) has also been studied in [1-9]. As far as we know the local convergence analysis of methods (1.2) and (1.3) has not been given. In the present paper, we study the local convergence of the Ulm’s-like method de ned for each n = 0; ; 2; 3; : : : by

equation

where An is an approximation of F’(xn). Notice that method (1.4) is inverse free, the computation of F0(xn) is not required and the method produces suc-cessive approximations {Bn} ≈ F’(x*)-1

In Section 2, we present the local convergence analysis of the method (1.4) and in Section 3, we present the numerical examples.

Local convergence analysis

The local convergence analysis of the method (1.4) is given in this section. Denote by U (v, ζ) the open and closed balls in B1, respectively, with center v ∈ B1 and of radius ζ>0.

Let w0 : [0,+ ∞] → [0,+ ∞] and w : [0,+ ∞] →[0,+ ∞] be continuous and nondecreasing functions satisfying w0 (0)= w(0)=0.

Let also q ∈ [0,1] be a parameter. Define functions ϕ and ψ on the interval [0,+ ∞] by

equation

and

equation

We have that ψ (0) = −1 and for sufficiently large 0 0 t ≥ t,ψ (t ) > 0 . By the intermediate value theorem equation ψ (t) = 0 has solutions in the interval (0, t0). Denote by the smallest such solution. Then, for each t ∈[0,ρ ]we have

0 ≤ψ (t) < 1. (2.1)

We need to show an auxiliary perturbation result for method (1.4).

LEMMA 2.1 Let equation be a continuously Frechet-differentiable operator. Suppose that there exist equation , continuous and nondecreasing functionsequation and equation such that for each x∈Ω,n = 0,1,2,.. and θ ∈[0,1]

equation

equation

equation

equation that for eachequation

equation

where equation

equation

and equation

where equation

Then, the following items hold

equation

equation

equation

And

equation

Proof we shall first show estimate (2.11) holds. Using (2.1), we have the identity

equationequation (2.14)

Then, by (2.4) and (2.14) we have that

equation

equation

which shows (2.10). Moreover, by (2.5), (2.6) and (2.10) we obtain that

equation

which shows the estimate (2.11). Furthermore, using (2.3), (2.4), (2.10), (2.11) and the definition of r0 we get that

equation (2.15)

equation

equation

It follows from (2.15) and the Banach lemma on invertible operators [1,4,6,11] that (2.12) and (2.13) hold.

REMARK 2.2 In earlier studies the Lipschitz condition [1-15]

equation (2.16)

is used which is stronger than our conditions (2.3) and (2.4). Notice also that sinceequation

equation (2.17)

and

equation (2.18)

where functions w1 is as function w but defined on Ω instead of Ω0.The ratio equation can be arbitrarily large [1,4,6]. Moreover, if (2.16) is used instead of (2.3) and (2.4) in the proof of Lemma 2.1, then the conclusions hold provided that r0 is replaced by r1 which is the smallest positive solution of equation

equation (2.19)

where equation it follows from (2.10), (2.17), (2.18), (2.19) that

equation

Furthermore, strict inequality holds in (2.20), if (2.17) or (2.18) hold as strict

Inequalities. Finally, estimates (2.11) and (2.12) are tighter than the corresponding ones (using (2.16)) given by

equation

Let λ be a parameter satisfying be a continuous and no decreasing function.

equation

equation

Moreover, define functions

equationequationequation

equation

equationequation

equation

Parameters equationand quadratic equatioequation Then, we haveequation

Denote byρ0 the smallest solution of equation f(t)=0 in (0,p) tthen, we have

that for each equation

equation

In view of the above inequality the preceding quadratic equation has a unique positive solution denoted by ρ+ and a negative solution. Define parameter γ by

equation

Then, we have that

equation

Notice that we also have that equation and equation

Next, we present the local convergence of method (1.4).

THEOREM 2.3 Under the hypotheses of Lemma 2.1 and with r0 given in (2.9) for λ ∈[0,1) further suppose there exists function 2 0 w :[0,r )→[0,+∞) continuous and no decreasing such that for each equation and

equation

equation for each equation

equation and

equation

where γ is given in (2.22). Then, sequence equation generated by the method (1.4) for equation is well defined , remains in * B(x ,γ ) and converges to x*

Proof. We have by hypothesis (2.25) that equation so

equation

is true for k = 0: Suppose that (2.27) is true for all integers smaller or equal to k: Using Lemma 2.1, we have the estimate

equation

equation

equation

In view of method (1.4) for n = k; we can write in turn that

equation

equation

equation

equation

equation

equation

equation

equation

equation

equation

By the definition of method (1.4), we have the estimate

equation

Then, by (2.32), (2.29) for n = k; we get in turn that

equation

equation

equation

equation

equation

equation

equation

which shows (2.27) for n = k +1: Then, using the induction hypotheses, (2.24), and the definition of γ

equation

equation

equation

where c = g[0,1], so equation

REMARK 2.4 (a) As noted in Remark 2.2 conditions (2.4) and (2.5) can be replaced by (2.24).

equation (2.36)

for each x∈Ω and θ ∈[0,1], where function ω3is as ω1:

We have that ω1 (t)≤ ω3 (t). Then, in view of Remark 2.2 and (2.24) the radii of convergence as well as the error bounds are improved under the new approach, since old approaches use only (2.36) with the exception of our approach in [2,5].

The results obtained here can be used for operators F satisfying autonomous differential equations [1,4,6,11] of the form

equation

Where equation is a continuous operator. Then, since F′(x*)= P(F(x*))= P(0), we can apply the results without actually knowing x* For example, let F(x) =ex-1. Then, we can choose P(x) = x + 1

(c) The local results obtained here can be used for projection methods such as the Arnoldi’s method, the generalized minimum residual method (GM-RES), the generalized conjugate method (GCR) for combined Newton/finite projection methods and in connection to the mesh independence principle can be used to develop the cheapest and most efficient mesh refinement strategies [1,4,6].

(d) Let L0, L, L1, L2, L3 be positive constants. Researchers, choose ω0(t)= L0t, ω(t)= Lt, ω1(t)= L1t, ω2(t)= L2t, and ω3(t)= L3t, Moreover, if we choose Ω0=Ω and L=L1 then, our results reduce to the ones given by where the second order of convergence was shown with the Lipschitz conditions given in non-affine invariant form. In Example 3.1, we shall show that the radii are extended and the upper bounds on ||xn- x*|| are tighter if we use ω0, ω, ω2 instead of using ω0 and ω we used in [5] or only ω3 as used in [2,7-15].

Numerical examples

Example 3.1 let equation Define function F on D for ω=(x,y,z)T by

equation

Then, the Frechet-derivative is defined by

equation

Notice that using the Lipschitz conditions, we get equationequation and equation where equation and equation Moreover, chooseequation to obtain equation

The parameters are equation

where the bar answers corresponding to the case when only ω3 is used in the derivation of the radii.

Example 3.2 Let

equation for natural integer

equation X and Y are equipped with the max-normequationequation The corresponding matrix norm is

equation

For equation On the interval [0; 1], we consider the following two point boundary value problem

equation (3.1)

[6,8,9,11]. To discretize the above equation, we divide the interval [0; 1] into m equal parts with length of each part: h=1/m and coordinate of each point: xi=I h with i=0,1,2,…,m. A second-order finite difference discretization of equation (3.1) results in the following set of nonlinear equations

equation (3.2)

Where equation For the above system-of-nonlinearequations, we provide the Frechet derivative

equation

We see that for

equation

equation where equation The parameters areequation

where the bar answers corresponding to the case when only ω3 is used in the derivation of the radii.

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